Topology

Definition

Topology is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

Related concepts

Abel Prize Abstract algebra Aleksandr Aleksandrov (mathematician)Alexander–Briggs notation Alexei Kitaev Algebra Algebraic geometry Algebraic number theory Algebraic topology Analysis Situs (paper)Analytic geometry Analytic number theory Andrey Kolmogorov Applied mathematics Arithmetic Arithmetic geometry Augustin-Louis Cauchy Banach fixed-point theorem Bernhard Riemann Betti number Boolean algebra (structure)Bundle (mathematics)CW complex Calabi–Yau manifold Calculus Cantor set Category theory Cesare Arzelà Characteristic classes Characterizations of the category of topological spaces Chern class Circle Circuit topology Classification theorem Clifford A. Pickover Clopen set Closed set Closure (mathematics)Cobordism Cohomology Colin Adams (mathematician)Combinatorial topology Combinatorics Commutative algebra Compact space Compactness Complex analysis Complex geometry Complex plane Compressive strength Computational complexity theory Computational geometry Computational mathematics Computer algebra Computer science Condensed matter physics Configuration space (physics)Conformal geometry Connected space Connectedness Contact mechanics Continuity (mathematics)Continuity (topology)Continuous function Continuum (topology)Control theory Covering space Cowlick Cross-cap Crumpling Curvature David Richeson David Thouless De Rham cohomology Deformation theory Dennis Sullivan Differentiable function Differentiable manifold Differential equation Differential geometry Differential topology Digital topology Diophantine geometry Discrete geometry Discrete mathematics Disentanglement puzzle Domain theory Duncan Haldane Edward Witten Electrophoresis Elementary algebra Encyclopedia of Mathematics Engineering mathematics Enrico Betti Equivariant topology Euclidean geometry Euclidean space Euler's Gem Euler's polyhedron formula Euler characteristic Eulerian path European Mathematical Society Family of sets Felix Hausdorff Fields Medal Figure-eight knot (mathematics)Finite geometry Foundations of mathematics Four-manifold Fourier series Fractal dimension Free group Function (mathematics)Functional analysis Fundamental group Future of mathematics Game theory General topology Genus (topology)Geometric topology Geometrization conjecture Geometry Geometry & Topology Georg Cantor Giulio Ascoli Glossary of mathematical symbols Gottfried Wilhelm Leibniz Graduate Texts in Mathematics Graph theory Greek language Grothendieck topology Group theory Hairy ball theorem Handle decomposition Harmonic analysis Hassler Whitney Hausdorff space Henri Poincaré Henry Segerman Heyting algebra History of mathematics History of the separation axioms Homeomorphic Homeomorphism Homological algebra Homology (mathematics)Homotopies Homotopy Homotopy equivalence Homotopy group Homotopy theory Hypercomplex analysis Informal mathematics Information theory Injective function Interior (topology)Invariance of domain Inverse image Jacques Hadamard James Munkres Johann Benedict Listing John L. Kelley Joint Jordan-Schönflies theorem Journal of Topology Kaliningrad Kazimierz Kuratowski Klein bottle Knot theory Large numbers Lattice (order)Lebesgue covering dimension Lemniscate Leonhard Euler Line (geometry)Linear algebra List of algebraic topology topics List of continuity-related mathematical topics List of examples in general topology List of films about mathematicians List of general topology topics List of geometric topology topics List of important publications in mathematics List of mathematicians List of publications in mathematics List of topology topics Lists of mathematics topics Local flatness Low-dimensional topology Ludwig Schläfli Manifold Materials science Mathematical Sciences Publishers Mathematical analysis Mathematical and theoretical biology Mathematical chemistry Mathematical economics Mathematical finance Mathematical geology Mathematical linguistics Mathematical logic Mathematical object Mathematical optimization Mathematical physics Mathematical psychology Mathematical sociology Mathematical statistics Mathematicians Mathematics Mathematics and art Mathematics education Maurice Fréchet Maxim Kontsevich Measure (mathematics)Mechanical engineering Metric (mathematics)Metric space Michael Artin Michael Kosterlitz Mikhail Lavrentyev Moduli spaces Molecules Motion planning Multilinear algebra Möbius strip Nature (journal)Neighborhood (mathematics)Nicolas Bourbaki Nobel Prize Number Number theory Numerical analysis Open set Operations research Orbit space Order theory Orientability Orientable manifold Persistent homology Peter Johnstone (mathematician)Philosophy of mathematics Physical cosmology Pi-system Plane (geometry)Plushie Poincaré conjecture Point-set topology Pointless topology Polyhedral complex Polyhedron Polyhedron formula Prime knot Probability theory Programming language semantics Projective plane Quantum Hall effect Quantum computing Quantum field theory R. J. Daverman Real analysis Real line Real number Real projective plane Recreational mathematics Riemannian curvature Robot Ronald Brown (mathematician)Ryszard Engelking Samson Abramsky Scientific journal Second-countable space Semidecidable Set-theoretic topology Set (mathematics)Set intersection Set theory Set union Seven Bridges of Königsberg Shape of the universe Sheaf (mathematics)Simon Donaldson Simplicial complex Smooth manifold Smooth structure Spacetime topology Sphere Statistics Steve Vickers (computer scientist)Stretch factor String theory Subset Surface (mathematics)Surface (topology)Surgery theory Surjective function Systems science The Geometry Center Theory of computation Timeline of mathematics Topography Topoisomer Topological Galois theory Topological data analysis Topological geometry Topological invariant Topological manifold Topological order Topological properties Topological property Topological quantum computer Topological quantum field theory Topological space Topology (disambiguation)Topology (structure)Topology glossary Torsion (mechanics)Torus Tychonoff's theorem Type theory Typology Uniform space Uniformization theorem Universal algebra Up to Urysohn's lemma Van Kampen's theorem Vaughan Jones Vector field Vector space Vito Volterra Wacław Sierpiński Winding number

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